let-x-gt-0-y-gt-0-z-gt-0-prove-that-x-2-yz-y-2-xz-z-2-xy-3-

Question Number 42492 by maxmathsup by imad last updated on 26/Aug/18

l e t x > 0, y > 0, z > 0 p r o v e t h a t \frac{x^{2}}{y z} + \frac{y^{2}}{x z} + \frac{z^{2}}{x y} ⩾ 3 .

Answered by tanmay.chaudhury50@gmail.com last updated on 26/Aug/18

\frac{x^{3}}{x y z} + \frac{y^{3}}{x y z} + \frac{z^{3}}{x y z} - 3

\frac{x^{3} + y^{3} + z^{3} - 3 x y z}{x y z}

\frac{(x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)}{x y z}

\frac{(x + y + z) {{(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2}}}{2 x y z}

+ v e (s i n c e x > 0 y > 0 z > 0)

s o \frac{x^{2}}{y z} + \frac{y^{2}}{x z} + \frac{z^{2}}{x y} > 3 p r o v e d

n o w i f (x = y = z

t h e n \frac{x^{2}}{y z} + \frac{y^{2}}{x z} + \frac{z^{2}}{x y} - 3 = 0

h e n c e \frac{x^{2}}{y z} + \frac{y^{2}}{z x} + \frac{z^{2}}{x y} ⩾ 3 p r o v e d

Commented by math khazana by abdo last updated on 27/Aug/18

c o r r e c t s i r t a n m a y t h a n k s .

Commented by tanmay.chaudhury50@gmail.com last updated on 27/Aug/18

i t s o k s i r

Answered by behi83417@gmail.com last updated on 26/Aug/18

\frac{x^{2}}{y z} + \frac{y^{2}}{x z} + \frac{z^{2}}{x y} ⩾ 3 \sqrt[3]{\frac{x^{2}}{y z} . \frac{y^{2}}{x z} . \frac{z^{2}}{x y}} = 3 \sqrt[3]{1} = 3 .